4.24.2005

Q&A: What are the lower bounds for European and American calls and puts with...

...exercise prices of $60, given that all options expire in 60 days, in which the current price of the underlying is $50 and the risk-free rate is 5%? What if the exercise price is $40 instead?

CppQuant Answer

  • Time to expiration (T) = 60/365 = 0.1644.
  • American Call: C(0) >= Max [0, S0 - X] = MAX[0, 50 - 60] = 0.
  • European Call: C(0) >= Max [0, S(0) - X / (1 + r)^T] = MAX[0, 50 - 60/(1 + 5%)^0.1644] = MAX[0, -9.95] = 0.
  • American Put: P(0) >= Max [0, X - S0] = MAX[0, 60 - 50) = 10.
  • European Put P(0) >= Max[0, X / (1 + r)^T - S(0)] = Max[0, 60/(1 + 5%)^0.1644 - 50] = MAX[0, 9.95) = 9.95.
  • the higher the exercise price, the lower the price of a call and the higher the price of a put. (try with excise price of $40)

Bonus Points

  • A Call option Payoff
    • min (European/American) = 0
      • American: C0 >= Max (0, S0 - X),
      • European: c(0) >= Max [0, S(0) - X / (1 + r)^T] . Construct a portfolio consisting of a long call and risk-free bond and a short position in the underlying asset. First we need the ability to buy and sell a risk-free bond with a face value equal to the exercise price and current value equal to the present value of the exercise price. We buy the European call and the risk-free bond and sell short (borrow the asset and sell it) the underlying asset. At expiration we shall buy back the asset.
    • max (European/American) = the underlying price
  • A Put option Payoff Boundary
    • min (European/American) = 0
      • American: P0 >= Max (0, X - S0)
      • European: p(0) >= Max[0, X / (1 + r)^T - S(0)] . Construct a portfolio consisting of a long put, a long position in the underlying, and the issuance of a zero-coupon bond. This combination produces a non-negative value at expiration so its current value must be non-negative. For this situation to occur, the put price has to be at least as much as the present value of the exercise price minus the underlying price.
    • max
      • American = the exercise price
      • European = PV of the exercise price: X/(1 + r)^t

Category: C++ Quant > Derivatives > Options

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